\documentclass{article}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{newunicodechar}
  \newunicodechar{⁻}{${}^{-}$}% Superscript minus
  \newunicodechar{²}{${}^{2}$}% Superscript two
  \newunicodechar{³}{${}^{3}$}% Superscript three

\pagestyle{empty}
\begin{document}
$(x_1, ..., x_n)$
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$(y_1, ..., y_n)$
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$y_i = sum_{j\le i} x_j$
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\[ //! M(h(X) \bmod G)= \sum_{i=0}^{d-1}(C[j] \cdot h(X^{p^j}))\bmod G). //! \]
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\[ CRT(vp,p,vq,q) = vp + [(vq-vp) * p^{-1}]_q * p, \]
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\[ M(h(X) mod G) = sum_{i=0}^{d-1} (C[j] mod G) * (h(X^{p^j}) mod G).\]
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$\mathbb{Z}_{p^r}[x]/G$
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$p^r$
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$ G$
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$\mathbb{Z}[X]/(G(x),p^r)$
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$\Phi_m(X)$
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$\mathbb{Z}_p[x]/\Phi(m)$
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$\mathbb{Z}_p[x]/G$
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$\Phi(m)$
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\end{document}
